
([1]}) When \(\lbrace \sigma _i\rbrace \)  is deterministic, there exists an absolute constant \(c\)  such that for any fixed \(\delta \in (0,1)\) , with probability at least \(1 - \delta \) .
\(*{\sum _{i=1}^n {\mathbf {x}}_i} \le c \sqrt{\log (2d/\delta )\sum _{i=1}^n \sigma _i^2 }\) 

([1]}) Suppose that the \(\lbrace \sigma _i\rbrace \)  sequence is random. Then, there exists an absolute constant \(c\)  such that for any fixed \(\delta \in (0,1)\)  and \(B > b > 0\) , with probability at least \(1 - \delta \) :
\(\mbox{either } \sum _{i=1}^n \sigma _i^2 \ge B \quad \mbox{ or } *{\sum _{i=1}^n {\mathbf {x}}_i} \le c \sqrt{\max \left\lbrace \sum _{i=1}^n \sigma _i^2,b \right\rbrace  \cdot (\log (2d/\delta ) + \log (\log (B/b)))}\) 


